Automorphism groups of homogeneous structures

This project is on the interface between model theory (mathematical logic), combinatorics, and permutation group theory. A countably infinite structure over a finite relational language (such as a graph) is homogeneous if any isomorphism between finite substructures extends to an automorphism of the whole structure (this means, roughly, that if two finite pieces look the same, then there is a symmetry of the whole structure taking one to the other). S. Thomas conjectured in 1996 that every such structure has finitely many `reducts', that is, the automorphism group has finitely many `closed' supergroups in the full symmetric group. This conjecture has received high attention over the last 15 years due to interactions with topological dynamics, with combinatorics (Ramsey theory), and with constraint satisfaction problems in theoretical computer science. The conjecture remains wide open but has been verified in some specific cases, and now seems accessible for wide classes of structures.

The main aim of this PhD project is to prove Thomas's conjecture for NIP homogeneous structures of finite thorn rank - these are model-theoretic tameness conditions of high current interest, and the class has been made accessible (whilst still being rich) due to recent advances of P. Simon. The planned approach is to combine Simon's structure theory with a Ramsey-theoretic method developed by Bodirsky and Pinsker. Initial goals are to classify the reducts of certain binary `circular' structures, and more generally of NIP rank 1 structures. The project will also examine the conjecture for certain NIP structures of infinite thorn rank, with a view to potential counterexamples.